Eames Walnut Stool Care  

I just polish mine with Orange Oil,
although you can use the Beeswax polish for a bit more protection.

Gunstock oil
You used to be able to order Gunstock oil from Herman Miller.
I called Herman Miller for the Home and no, they don't sell it anymore, but they said that some woodworking stores and lots of firearms stores still sell Gunstock oil and that if you can find it, it's the same animal.
Also, they suggested as a backup, clear linseed oil is the next best choice.

Tru-Oil
I found Birchwood Casey "Tru-Oil" (Gunstock finish) at the woodcraft.com website.
It's an 8oz. bottle for $10.99. Looks like the perfect product.
I'm going to buy a bottle myself.

Thanks for the info. Do you...
Thanks for the info. Do you know how often one should oil the stool and how to do it properly?

i have one, and just ...
i have one, and just polish it just like a piece of furniture, Is there something that you are suppose to do? i have never heard of doing anything special to them.

Unless
the product advises otherwise, oils are simply wiped on and wiped off. If the wood is dry-looking and "thirsty," it will take more oil and you can leave it on a bit before wiping off the excess. But on a well-finished piece, it's more of a freshener than anything. . .

Oil once or twice a year
I would oil it once or twice a year.
Rest of the time, if needed, I'd wipe it off with a soft cloth.

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Caranuba is the ingredient in the better wood protective products,I once used it on a pair of leather shoes still have them.This feeding wood with oil... in most cases it goops up a surface but its easy so alot of people really go in for that...